11 x1 t15 05 polynomial results (2012)

Polynomial Results

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

4 3 22 2 2 3 2 5 2 10

0P

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

4 3 22 2 2 3 2 5 2 10

0P

1, 2 are zeros of ( ) and 1 2 is a factorP x x x

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

4 3 22 2 2 3 2 5 2 10

0P

1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

4 3 22 2 2 3 2 5 2 10

0P

1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x

2 2 x x

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

4 3 22 2 2 3 2 5 2 10

0P

1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x

2 2 x x 2x

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

4 3 22 2 2 3 2 5 2 10

0P

1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x

2 2 x x 2x 5

Polynomial Results1. If P(x) has k distinct real zeros, , then;

is a factor of P(x).kaaaa ,,,, 321

kaxaxaxax 321

e.g. Show that 1 and –2 are zeros of and hence factorise P(x).

4 3 2( ) 3 5 10P x x x x x

4 3 21 1 1 3 1 5 1 100

P

4 3 22 2 2 3 2 5 2 10

0P

1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x

2 2 x x 2x 5

21 2 5x x x

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

3. A polynomial of degree n cannot have more than n distinct real zeros

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

3. A polynomial of degree n cannot have more than n distinct real zeros

e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;

a) a possible polynomial

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

3. A polynomial of degree n cannot have more than n distinct real zeros

e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;

a) a possible polynomial

2( ) 2 7P x x x

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

3. A polynomial of degree n cannot have more than n distinct real zeros

e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;

a) a possible polynomial

2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

3. A polynomial of degree n cannot have more than n distinct real zeros

e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;

a) a possible polynomial

2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7

k

- k is a real number

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

3. A polynomial of degree n cannot have more than n distinct real zeros

e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;

a) a possible polynomial

2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7

k

- k is a real number

b) a monic polynomial of degree 3.

2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321

naaaa ,,,, 321

3. A polynomial of degree n cannot have more than n distinct real zeros

e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;

a) a possible polynomial

2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7

k

- k is a real number

b) a monic polynomial of degree 3.

2( ) 2 7P x x x

c) A monic polynomial of degree 4

c) A monic polynomial of degree 4

2( ) 2 7P x x x

c) A monic polynomial of degree 4

2( ) 2 7P x x x x a

where 2 or 7a

d) a polynomial of degree 5.

c) A monic polynomial of degree 4

2( ) 2 7P x x x x a

where 2 or 7a

d) a polynomial of degree 5.

2( ) 2 7P x x x

c) A monic polynomial of degree 4

2( ) 2 7P x x x x a

where 2 or 7a

d) a polynomial of degree 5.

2( ) 2 7P x x x ( )Q x

where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7

c) A monic polynomial of degree 4

2( ) 2 7P x x x x a

where 2 or 7a

d) a polynomial of degree 5.

2( ) 2 7P x x x ( )Q x

where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7

k

- k is a real number

4. If P(x) has degree n and has more than n real zeros, then P(x) is the zero polynomial. i.e. P(x) = 0 for all values of x

c) A monic polynomial of degree 4

2( ) 2 7P x x x x a

where 2 or 7a

d) a polynomial of degree 5.

2( ) 2 7P x x x ( )Q x

where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7

k

- k is a real number

Exercise 4E; 1, 3bd, 4a, 5b, 6a, 8, 12ad